Discrete Algebraic Methods

Fig. 8.5. Two different graphical realizations of the Whitehead automorphism W(a,{b},{c},{a,d}) .

Given a graphical realization (Γ, Φ) and a word u ∈ (X̃)∗, we may interpret φ(u) as the word φΓ (u) in E ∗ or as an element of the free group F (E+) or by virtue of λ as an element of F (Y). However, there is no risk of confusion.

The trivial homomorphism has a graphic realization by a graph with two points 1 and 2, which are connected in the direction from 1 to 2 by |X| + 1 different arcs and all of them have label y, where y is some fixed letter in the nonempty set Y.

The identity map id: F(X) → F (X) is realized by a so-called rose. This is a ...